[Math.QA] 24 Jan 2003 Ihagbacsprrushwvr Hr R Nsm Sense Some in Are There However, Supergroups Algebraic with En Anyte Pwt Hsclproblems

arXiv:math/0111113v2 [math.QA] 24 Jan 2003 ihagbacsprrushwvr hr r nsm sense some in are there however, supergroups algebraic with en anyte pwt hsclproblems. physical amo with [Ma1] up Manin and tied [Ko] mainly Kostant being [Le] Leites by later and [Be] topics. ntecmuaiecs.I shnepsil odfieaquant it. a to ca define associated super superalgebra to non Hopf possible the commutative to the hence of analogy is complete It in ca supergroup this algebraic case. superalg in Hopf commutative is commutative the points a of in supergroup functor affine of an view to of associate point the reason this nw rmamr oeainl on fview. of point “operational” a more develop a Deligne notions from Supersymmetry” supersymmetry known on on view “Notes of point 1999 categorical the In superstring. hog h uco fpit n erlt hmt h te defi other the to supergroups them algebraic relate the detail we in and study points of functor the through fatmrhsso superspace a of automorphisms of algebra supercommutative tutrlsefo ucin ntemnfl.Ti ha is sheaf addi a This an is to superalgebra manifold. a refers the where super on superalgebras, adjective commutative functions the of and sheaf manifold structural usual an of tutr fteagbarpeetn h ucoso ons tth At points. of functors the GL representing algebra the of structure netgto upre yteUiest fBlga fun Bologna, of University the by supported Investigation * ( m h ahmtclfudtoso uegoer eeli in laid were supergeometry of foundations mathematical The Introduction 1. e teto otesbetcm ae ihtesuyo qua of study the with later came subject the to attention new A ntecretdfiiin fspraiod h onso s a of points the supermanifold, of definitions current the In Abstract. | n ) oehrwt t ocino utbeqatmsae accord spaces quantum suitable on coaction its with together egv h entoso ffieagbacsprait n ffiea affine and supervariety algebraic affine of definitions the give We iatmnod aeaia nvria iBologna di Universita’ Matematica, di Dipartimento izaPraSnDnt ,416Blga Italy Bologna, 40126 5, Donato San Porta Piazza NAGBACSPRRUSAND SUPERGROUPS ALGEBRAIC ON UNU DEFORMATIONS QUANTUM A the A -al fi[email protected] e-mail: A onso uegopcnb iwda eti subset certain a as viewed be can supergroup a of points GL m | n .FIORESI* R. ( where m | n ) 1 and A sagvncmuaiesprler.For superalgebra. commutative given a is SL ( m | Z n n egv loteqatzto of quantization the also give we end e 2 ) iin rsn nteltrtr.We literature. the in present nitions gae ler.We dealing When algebra. -graded n ieepiil h ofalgebra Hopf the explicitly give and doiial ypyiit and physicists by originally ed true n oMnnsphilosophy. Manin’s to ing ems sfl talw to allows It useful. most se e twl eadeformation a be will it se: gmn tes t origins its others, many ng batesm a tdoes it way same the ebra sue ob ha of sheaf a be to assumed dMra D]gv a give [DM] Morgan nd sfrslce research selected for ds praiodaepoints are upermanifold inlsrcueo the on structure tional ons nfc,freach for fact, In points. mdfraino an of deformation um h 0 yBrznin Berezin by 60s the gbacsupergroup lgebraic tmfilsand fields ntum The quantum supergroup GL(m|n) was first constructed by Manin [Ma3] together with its coactions on suitable quantum superspaces. The construction of its Hopf superalgebra structure is in some sense implicit. In a subsequent work, Lyubashenko and Sudbery [LS] provided quantum deformations of supergroups of GL(m|n) type using the universal R-matrix formalism. The same formalism is also used in [P] where more explicit formulas are given. In both works however the ring and its coalgebra structure do not appear explicitly, since the calculation was too involved using the R-matrix approach. In the present paper we present a definition of affine algebraic supervariety and supergroup that is basically equivalent to the one of Manin [Ma1] and then we use this point of view to give a quantum deformation of the supergroup GL(m|n). The main result of the paper consists in giving explicitly the coalgebra structures for the Hopf superalgebras associated to the supergroup GL(m|n) and its quantization kq[GL(m|n)] obtained according to the Manin phylosophy, that is together with coactions on suitable quantum spaces. The explicit forms of the comultiplication for GL(m|n) and its quantization kq[GL(m|n)] are non trivial, they rely heavily on the presence of the nilpotents of the superalgebras and do not appear in any other work. These results were announced in the proceeding [Fi] where they appeared without proof. The organization of this paper is as follows. In §2 we introduce the notion of affine supervariety and affine supergroup using the functor of points. These two definitions turn out to be basically equivalent to the definitions that one finds in the literature ([De], [Ma1] among many others). In §3 we write explicitly the Hopf algebra structure of the Hopf superalgebra associated to the supergroups GL(m|n) and SL(m|n), where with GL(m|n) we intend the supergroup whose A points are the group of automorphisms of the superspace Am|n and with SL(m|n) the subsupergroup of GL(m|n) of automorphisms with Berezinian equal to 1, with A commutative superalgebra.

In §4 we construct the non commutative Hopf superalgebra kq[GL(m|n)] deformation in the quantum group sense of the Hopf superalgebra associated to GL(m|n). We also see that kq[GL(m|n)] admits coactions on suitable quantum superspaces.

The author wishes to thank Prof. V. S. Varadarajan, Prof. I. Dimitrov, Prof. A. Vistoli, Prof. A. Sudbery, Prof. T. Lenagan and Prof. M. A. Lledo for helpful comments.

2. Preliminaries on algebraic supervarieties and supergroups

Let k be an algebraically closed ﬁeld. All algebras and superalgebras have to be intended over k unless otherwise speciﬁed. Given a superalgebra A we will denote with A0 A the even part, with A1 the odd part and with Iodd the ideal generated by the odd part.

2 A superalgebra is said to be commutative (or supercommutative) if

xy =(−1)p(x)p(y)yx, for all homogeneous x,y

where p denotes the parity of an homogeneous element (p(x)=0if x ∈ A0, p(x)=1 if

x ∈ A1). In this section all superalgebras are assumed to be commutative. Let’s denote with A the category of affine superalgebras that is commutative superalgebras such that, modulo the ideal generated by their odd part, they are affine algebras (an affine algebra is a finitely generated reduced commutative algebra). Definition (2.1). Define affine algebraic supervariety over k a representable functor V from the category A of affine superalgebras to the category S of sets. Let’s call k[V ] the commutative k-superalgebra representing the functor V ,

V (A) = Homk−superalg(k[V ],A), A ∈A.

We will call V (A) the A-points of the variety V . A morphism of affine supervarieties is identified with a morphism between the representing objects, that is a morphism of affine superalgebras.

We also deﬁne the functor Vred associated to V from the category Ac of aﬃne k- algebras to the category of sets:

k[V ] Vred(Ac) = Homk−alg(k[V ]/Iodd ,Ac), Ac ∈Ac.

Vred is an affine algebraic variety and it is called the reduced variety associated to V . If the algebra k[V ] representing the functor V has the additional structure of a commutative Hopf superalgebra, we say that V is an affine algebraic supergroup. (For the definition and main properties of Hopf algebras see [Mo]). Remarks (2.2). 1. Let G be an affine algebraic supergroup in the sense of (2.1). As in the classical setting, the condition k[G] being a commutative Hopf superalgebra makes the functor group valued, that is the product of two morphisms is still a morphism.

In fact let A be a commutative superalgebra and let x,y ∈ Homk−superalg(k[G],A) be two points of G(A). The product of x and y is deﬁned as:

x · y =def mA · x ⊗ y · ∆

where mA is the multiplication in A and ∆ the comultiplication in k[G]. One can directly check that x · y ∈ Homk−superalg(k[G],A), that is:

(x · y)(ab)=(x · y)(a)(x · y)(b)

3 This is an important diﬀerence with the quantum case that will be treated in §4. The non commutativity of the Hopf algebra in the quantum setting does not allow to multiply morphisms(=points). In fact in the quantum (super)group setting the product of two morphisms is not in general a morphism. For more details see [Ma4] pg 13.

2. Let V be an aﬃne algebraic supervariety as deﬁned in (2.1). Let k0 ⊂ k be a

subﬁeld of k. We say that V is a k0-variety if there exists a k0-superalgebra k0[V ] such ∼ that k[V ] = k0[V ] ⊗k0 k and

V (A) = Homk0−superalg(k0[V ],A) = Homk−superalg(k[V ],A), A ∈A.

We obtain a functor that we still denote by V from the category Ak0 of aﬃne k0-superalgebras to the category of sets:

V (Ak0 ) = Homk0−superalg(k0[V ],Ak0 ), A ∈Ak0 .

This allows to consider rationality questions on a supervariety. We will not pursue this further in the present work. Examples (2.3). 1. The k-points of an affine supervariety V correspond to the affine variety defined

over k whose functor of points is Vred. 2. Let A be a commutative superalgebra. Let M(m|n)(A) be the linear endomor- phisms of the superspace Am|n (see [De] pg. 53):

a11 ... a1,m α1,m+1 ... α1,m+n  . . . .  . . . .  am, ... am,m αm,m ... αm,m n   1 +1 +   αm , ... αm ,m am ,m ... am ,m n   +1 1 +1 +1 +1 +1 +   . . . .   . . . .     α ... α a ... a   m+n,1 m+n,m m+n,m+1 m+n,m+n  aij ∈ A0, αkl ∈ A1, 1 ≤ i, j ≤ m or m +1 ≤ i, j ≤ m + n, 1 ≤ k ≤ m, m +1 ≤ l ≤ m + n or m +1 ≤ k ≤ m + n, 1 ≤ l ≤ m.

This is an aﬃne supervariety represented by the commutative superalgebra: k[M(m|n)] = k[xij,ξkl] where xij’s and ξkl’s are respectively even and odd variables with 1 ≤ i, j ≤ m or m +1 ≤ i, j ≤ m + n,1 ≤ k ≤ m, m +1 ≤ l ≤ m + n or m +1 ≤ k ≤ m + n, 1 ≤ l ≤ m.

Observe that Mred = M(m) × M(n) where M(l) is the functor corresponding to the aﬃne variety of l × l matrices.

4 We now would like to give an equivalent point of view and define again the category of affine supervarieties and affine supergroups. (See [Ma1], [De], [Be]).

Definition (2.4). Let Vred be an affine algebraic variety defined over k and OVred the structural sheaf of Vred. Define affine algebraic supervariety V the couple (Vred,OV ) where

OV is a sheaf of aﬃne superalgebras such that its stalk is local and OV /IV is isomorphic

to OVred , where IV is the sheaf of ideals generated by the nilpotent elements. We want to show that this definition is equivalent to the one given previously. Clearly if we have an affine supervariety according to the definition (2.4), we have a superalgebra associated to it, namely the global sections of the sheaf OV . This means that we immediately have the functor of points associated to it, hence a supervariety according to the definition (2.1). Conversely assume we have a functor of points V and a commutative superalgebra k[V ] to which it is associated (see definition (2.1)). We need to show that it gives rise to a sheaf of superalgebras on the affine variety Vred. Let’s look at the maps:

α β k[V ] k[V ]0֒→k[V ]−→k[V ]/Iodd k[V ] where Iodd is the ideal generated by the nilpotent elements in k[V ]. Observe that we have a surjective map γ = β · α. whose kernel consists of the nilpotent elements of k[V ]0. This induces a map k[V ] Speck[V ]/Iodd −→ Speck[V ]0 that is an isomorphism since the kernel of γ consists of nilpotents. Let’s now view k[V ] k[V ] as an k[V ]0-module. This allows us to build a sheaf on Speck[V ]0 = Speck[V ]/Iodd of k[V ]0-modules, where the stalk coincides with the localization of k[V ] into the maximal ideals of k[V ]0. So we have obtained from a commutative superalgebra k[V ] a sheaf of k[V ] superalgebras on Speck[V ]/Iodd which corresponds to the aﬃne variety Vred, whose global sections coincide with k[V ]. The next section will be devoted to construct in detail examples of aﬃne supergroups.

3. The affine supergroups GL(m|n) and SL(m|n) In this section we intend to give explicitly the supergroup structure for the supergroup functors GL(m|n) and SL(m|n). For any A ∈A let’s define GL(m|n)(A) as the group of automorphisms of the superspace Am|n (see [De] pg 59). Define also SL(m|n)(A) as the subset of GL(m|n) of automorphisms with berezinian equal to 1. The berezinian of the matrix: G Γ 11 12 ∈ SL(m|n)(A), Γ21 G22

5 (G11, G22 are m × m, n × n invertible matrices of even elements, Γ12, Γ21 are m × n, n × m matrices of odd elements) is deﬁned as:

−1 −1 Ber = det(G22) det(G11 − Γ12G22 Γ21).

(See [Be] ch. 4, [De] pg 59 for more details).

From section 2 we know that we need to give Hopf commutative superalgebras k[GL(m|n)] and k[SL(m|n)] such that for each commutative superalgebra A,

GL(m|n)(A) = Homk−superalg(k[GL(m|n)],A)

SL(m|n)(A) = Homk−superalg(k[SL(m|n)],A).

We start by deﬁning the supercommutative algebras k[GL(m|n)] and k[SL(m|n)], then we will give explicitly its coalgebra and Hopf algebra structure.

Let xij for 1 ≤ i, j ≤ m or m +1 ≤ i, j ≤ m + n be even variables and ξkl for 1 ≤ k ≤ m, m +1 ≤ l ≤ m + n or m +1 ≤ k ≤ m + n, 1 ≤ l ≤ m be odd variables.

Denote by X11, X22, Ξ12, Ξ21 the following matrices of indeterminates:

X11 =(xij)1≤i,j≤m,X22 =(xij)m+1≤i,j≤m+n,

Ξ12 =(ξkl)1≤k≤m,m+1≤l≤m+n, Ξ21 =(ξkl)m+1≤k≤m+n,1≤l≤m.

Deﬁnition (3.1).

1...m−1 m+1...m+n−1 k[GL(m|n)] =def k[xij,ξkl,d1...m ,dm+1...m+n ]

1...m−1 m+1...m+n−1 k[xij ,ξkl,d1...m ,dm+1...m+n ] k[SL(m|n)] =def ( det(S22(X22))det(X11 − Ξ12S22(X22)Ξ21) − 1)

1...m−1 m+1...m+n−1 where d1...m , dm+1...m+n are even variables such that

1...m−1 1...m m+1...m+n−1 m+1...m+n d1...m d1...m =1, dm+1...m+n dm+1...m+n =1.

1...m m+1...m+n with d1...m = det(X11), dm+1...m+n = det(X22). 1...m−1 t 1...m−t To simplify the notation we will also write (d1...m ) as d1...m for t a positive integer.

i−j 11 1...m−1 S11(xij) =def (−1) Aji d1...m , 1 ≤ i, j ≤ m

i−j 22 m+1...m+n−1 S22(xij) =def (−1) Aji dm+1...m+n , m +1 ≤ i, j ≤ m + n

6 11 22 th where Aji and Aji denote the determinants of the minors obtained by suppressing the j th row and i column in X11 and X22 respectively.

Regarding S11(X11) as a matrix of indeterminates that is (S11(X11))ij = S11(xij), 1 ≤ i, j ≤ m, we have that:

S11(X11)X11 = X11S11(X11) = Im

Similarly:

S22(X22)X22 = X22S22(X22) = In

where Im and In denote the identity matrix of order m and n respectively. The expression:

Ber = det(S22(X22))det(X11 − Ξ12S22(X22)Ξ21)

is called the Berezinian function. (See [Be] ch. 3 for more details). Proposition (3.2). In the ring k[GL(m|n)] the Berezinian function is invertible. Proof. Let’s write Ber as:

Ber = det(S22(X22)det(Im − Ξ12S22(X22)Ξ21S11(X11))det(X11).

It is enough to prove that det(Im − Ξ12S22(X22)Ξ21S11(X11)) is invertible. To simplify the

notation let’s call A = Ξ12S22(X22), B = Ξ21S11(X11). We now prove that Im − AB is invertible (as matrix of indeterminates). It’s inverse is given by:

2 mn+1 Im +(AB)+(AB) + ... +(AB)

In fact if one multiplies this matrix with Im − AB, one obtains a telescopic sum with last term (AB)m+n. Each element of A and B is of degree 1 in the odd indeterminates hence (AB)mn+1 is of odd degree 2mn + 2 hence zero. We now proceed to give the coalgebra structure. We need to give the comultiplication ∆ and counit ǫ for all the generators and verify that they are well deﬁned. In order make the formulas more readable we introduce the notation:

xij if 1 ≤ i, j ≤ m or m +1 ≤ i, j ≤ m + n aij =def   ξij if 1 ≤ i ≤ m, m +1 ≤ j ≤ m + n or m +1 ≤ i ≤ m + n, 1 ≤ j ≤ m  1...m m+1...m+n Observation (3.3). Let’s compute ∆(d1...m) and ∆(dm+1...m+n). ∆(d1...m) = (−1)l(σ)∆(a ) ... ∆(a ) = 1...m σ∈Sm 1,σ(1) m,σ(m) P = (−1)l(σ)a ...a ⊗ a ...a = m r σ∈Sm,1≤k1,...,km≤m+n 1,k1 m,km k1,σ(1) km,σ(m) i=1 1 P P 7 with r = (−1)l(σ)a ...a ⊗ a ...a i σ∈Sm,(k1,...,km)∈ρi 1,k1 m,km k1,σ(1) km,σ(m) P ρi = {(k1,...,km)|1 ≤ k1,...,km ≤ m + n, |{k1,...,km}∩{1 ...m}| = m − i} Similarly one can write: n m+1...m+n ∆(dm+1...m+n) = si Xi=1 with s = (−1)l(θ)a ...a ⊗ a ...a i θ∈Sn,(l1,...,ln)∈σi 1,l1 n,ln l1,θ(1) ln,θ(n) P σi = {(l1,...,ln)|1 ≤ l1,...,ln ≤ m + n, |{l1,...,ln}∩{m +1 ...m + n}| = n − i}.

Proposition (3.4). k[GL(m|n)] and k[SL(m|n)] are bialgebras with comultiplication:

∆(aij) = aik ⊗ akl P 1...m−1 2mn+2 i−1 1...m−i 1...m−i 1...m 1...m 1...m i−1 ∆(d1...m ) = i=1 (−1) d1...m ⊗ d1...m (∆(d1...m) − d1...m ⊗ d1...m) P

m+1...m+n−1 2mn+2 i−1 m+1...m+n−i m+1...m+n−i ∆(dm+1...m+n ) = i=1 (−1) dm+1...m+n ⊗ dm+1...m+n P m+1...m+n m+1...m+n m+1...m+n i−1 (∆(dm+1...m+n) − dm+1...m+n ⊗ dm+1...m+n) and counit:

1...m−1 m+1...m+n−1 ǫ(aij) = δij, ǫ(d1...m )=1, ǫ(dm+1...m+n )=1.

Proof. One can directly check that these maps are well deﬁned with respect to the commutation relations among even and odd elements. For k[SL(m|n)]: Ber is a grouplike element, that is ∆(Ber) = Ber ⊗ Ber. A proof of this in the quantum case is available in [LS], [P], it clearly applies also to the non quantum case, that is when q = 1. Hence we have immediately:

∆(Ber − 1)=(Ber − 1) ⊗ Ber +1 ⊗ (Ber − 1)

We now need to check (for both algebras the computation is the same) that:

1...m−1 1...m (∆(d1...m ))(∆(d1...m))=1 ⊗ 1

m+1...m+n−1 m+1...m+n (∆(dm+1...m+n ))(∆(dm+1...m+n))=1 ⊗ 1

8 By observation (3.3) we have that:

1...m m+1...m+n ∆(d1...m) = r0 + ... + rm, ∆(dm+1...m+n) = s0 + ... + sn,

1...m−1 2mn+2 i−1 −i i−1 ∆(d1...m ) = i=1 (−1) r0 (r1 + ... + rm) P m+1...m+n−1 2mn+2 i−1 −i i−1 ∆(dm+1...m+n ) = i=1 (−1) s0 (s1 + ... + sn) P 1...m 1...m m+1...m+n m+1...m+n Notice that r0 = d1...m ⊗ d1...m and s0 = dm+1...m+n ⊗ dm+1...m+n. So:

1...m 1...m−1 −1 −2 (∆(d1...m))(∆(d1...m ))=[r0 +(r1 + ... + rm)][r0 − r0 (r1 + ... + rm) + ...]

−2 2 1 ⊗ 1+(r1 + ...rm)r0 − r0(r1 + ...rm) − r0 (r1 + ... + rm) + ... (∗)

We obtain a telescopic sum. The generic term is given by:

i−1 −i i−1 g =[r0 +(r1 + ... + rm)][... +(−1) r0 (r1 + ... + rm) +

i −i−1 i +(−1) r0 (r1 + ... + rm) + ...] =

i−1 −i+1 i−1 i−1 −i i = ... +(−1) r0 (r1 + ... + rm) +(−1) r0 (r1 + ... + rm) +

i −i i i −i+1 i+1 +(−1) r0 (r1 + ... + rm) +(−1) r0 (r1 + ... + rm) + ...

Notice that the second and third term cancel out. The last term in the sum (∗) is given by:

2mn+1 −2mn−2 2mn+1 (−1) r0 (r1 + ... + rm) =

(−1)2mn+1r−2mn−2 r ...r . 0 1≤i1,...,i2mn+1≤m i1 i2mn+1 P But

ri1 ...ri2mn+1 =0

since it contains the product of 2mn + 1 odd indeterminates (each ri for i > 1 contains at least one odd indeterminate). Hence we have the result. The check: m+1...m+n m+1...m+n−1 ∆(dm+1...m+n)∆(dm+1...m+n )=1 ⊗ 1 is done in the same way. Finally one directly checks that ǫ is a counit.

Remarks (3.5).

9 1. For n = m = 1 we have:

1 −1 −1 −1 −1 −2 −2 ∆((d1) )=∆(x11 ) = x11 ⊗ x11 − x11 ⊗ x11 (ξ12 ⊗ ξ21)

2 −1 −1 −1 −1 −2 −2 ∆((d2) )=∆(x22 ) = x22 ⊗ x22 − x22 ⊗ x22 (ξ21 ⊗ ξ12) that gives precisely the formulas in example (2.3)(3).

2. Notice that if one replaces the ξkl with commuting coordinates, k[GL(m|n)] and k[SL(m|n)] are not Hopf algebras. This comes from the fact that the product of two m+n by m + n matrices whose diagonal m × m and n × n blocks are invertible is not a matrix of the same type. The coalgebra and Hopf algebra structures use in an essential way the supercommutativity (i.e. the presence of nilpotents).

Let’s now deﬁne the antipode S.

Let B =(bij)m+1≤i≤m+n,1≤j≤m, C =(ckl)1≤k≤m,m+1≤l≤m+n be the following matrices of even elements: B = X11 − Ξ12S22(X22)Ξ21

C = X22 − Ξ21S11(X11)Ξ12 Deﬁne the matrices:

i−j B −1 S1(B)ij = S1(bij) =def (−1) Ajidet(X11 − Ξ12S22(X22)Ξ21)

l−k C −1 S2(C)kl = S2(ckl) =def (−1) Alkdet(X22 − Ξ21S11(X11)Ξ12)

B C th where Aji and Alk are the determinants of the minors obtained by suppressing the j row and ith column in B and the lth row and kth column in C respectively.

Remark (3.6). The determinants that appear in the deﬁnition of S1 and S2 are invertible in k[GL(m|n)]. The fact that the determinant det(X11 − Ξ12S22(X22)Ξ21) is invertible is contained in the proof of Proposition (3.2). The other determinant can be proven invertible in the same way. These determinants are also invertible in k[SL(m|n)]. In fact can be easily seen in k[SL(m|n)] by observing that since Ber = 1 and since:

−1 Ber = det(X11)det(X22 − Ξ21S11(X11)Ξ12)

(see [P]), we have:

−1 m+1...m+n det(X11 − Ξ12S22(X22)Ξ21) = det(S22(X22)) = dm+1...m+n

1...m det(X22 − Ξ21S11(X11)Ξ12) = det((X11)) = d1...m

10 Proposition (3.7). k[GL(m|n)] and k[SL(m|n)] are Hopf algebras with antipode S:

X Ξ S 11 12 = Ξ21 X22

S1(X11 − Ξ12S22(X22)Ξ21) −S11(X11)Ξ12S2(X22 − Ξ21S11(X11)Ξ12) −S22(X22)Ξ21S1(X11 − Ξ12S22(X22)Ξ21) S2(X22 − Ξ21S11(X11)Ξ12)

1...m−1 m+1...m+n m+1...m+n−1 1...m S(d1...m ) = dm+1...m+n S(dm+1...m+n ) = d1...m Proof. One can check directly that this map is well deﬁned and that is an antipode (See [Be]). Proposition (3.8). Let A be a commutative superalgebra. m|n 1. Homk−superalg(k[GL(m|n)],A) is the group of automorphisms of A . m|n 2. Homk−superalg(k[SL(m|n)],A) is the group of automorphisms of A with berezinian 1. k[GL(m|n)] and k[SL(m|n)] are the representing objects for the functors GL(m|n) and SL(m|n) respectively. Proof. Immediate.

4. The quantum GL(m|n) A quantum group is an Hopf algebra which is in general neither commutative nor cocommutative. According to this philosophy we can define a quantum supergroup in the same way. Definition (4.1). Let A be a commutative (super)algebra over k. A formal defor- −1 mation of A is a non commutative (super)algebra Aq over kq = k[q, q ], q being an (even) indeterminate, such that Aq/(q − 1) =∼ A. If A in addition is an Hopf superalgebra we will refer to such deformation as a quantum (super)group. We will call quantum GL(m|n) a formal deformation of the supercommutative Hopf algebra k[GL(m|n)] associated to the affine algebraic group GL(m|n) (see §3). We cannot define quantum groups using the functor of points as we did for supergroups. This is a consequence of the non commutativity of the Hopf algebra associated to the functor of points (see remark (2.2)(1)). We will also show that it is possible to give a deformation of the Hopf algebra k[GL(m|n)] in such a way that natural coactions on quantum superspaces are preserved.

In order to define the deformed algebra kq[GL(m|n)] we need first to define the Manin matrix superalgebra kq[M(m|n)] introduced by Manin in [Ma3] and to compute the commutation rules between the generators of kq[M(m|n)] and certain quantum determinants.

11 Deﬁnition (4.2). Deﬁne the following superalgebra ([Ma3]):

kq[M(m|n)] =def kq /IM

where kq < xij,ξkl > denotes the free algebra over kq generated by the even variables

xij for 1 ≤ i, j ≤ m or m +1 ≤ i, j ≤ m + n and by the odd variables ξkl for 1 ≤ k ≤ 2 m, m +1 ≤ l ≤ m + n or m +1 ≤ k ≤ m + n, 1 ≤ l ≤ m satisfying the relation: ξkl = 0. Let’s denote as before:

xij if 1 ≤ i, j ≤ m or m +1 ≤ i, j ≤ m + n aij =   ξij if 1 ≤ i ≤ m, m +1 ≤ j ≤ m + n or m +1 ≤ i ≤ m + n, 1 ≤ j ≤ m  The ideal IM is generated by the relations ([Ma3]):

p(i)+1 π(aij )π(a ) (−1) aijail =(−1) il q ailaij, j

p(j)+1 π(aij )π(a ) (−1) aijakj =(−1) kj q akjaij, i

π(aij )π(akl) aijakl =(−1) aklaij, il or i>k,j

π(aij )π(akl) aijakl − (−1) aklaij =

p(j)p(k)+p(j)p(l)+p(k)p(l) −1 π(a )π(a ) (−1) (q − (−1) il jk q)ajkail i

where p(i)=0if1 ≤ i ≤ m, p(i) = 1 otherwise and π(aij) denotes the parity of aij. Notice that for q = 1 this gives us the superalgebra defined in the example (2.3)(2) representing the functor M(m|n). m|n Definition (4.3). Define the quantum superspace kq as the ring generated over kq by the even elements x1 ...xm and the odd elements ξ1 ...ξn subject to the relations [Ma3]: −1 xixj − q xjxi, 1 ≤ i

−1 xiξk − q ξkxi, 1 ≤ i ≤ m, m +1 ≤ k ≤ n + m,

2 −1 ξk, ξkξl + q ξlξk, m +1 ≤ k

yiyj − qyjyi, 1 ≤ i

yiηk − qηkyi, 1 ≤ i ≤ m, m +1 ≤ k ≤ n + m,

2 ηk, ηkηl + qηlηk, m +1 ≤ k

12 Observation (4.4). The superalgebra kq[M(m|n)] admits a bialgebra structure with comultiplication and counit given by:

∆(aij) = aik ⊗ akj ǫ(aij) = δij P m|n m|n ∗ and a coaction on the quantum spaces kq and (kq ) ([Ma3]).

Observation (4.5). Let’s examine some of the relations that generate the ideal IM . For 1 ≤ i,j,k,l ≤ m:

−1 −1 xijxil = q xilxij , j

xij xkl = xklxij , il or i>k,j

−1 xijxkl − xklxij =(q − q)xilxkj, i

xijxil = qxilxij , j

xijxkl = xklxij, il or i>k,j

−1 xijxkl − xklxij =(q − q)xilxkj, i

These relations are the usual Manin relations where q is replaced with q−1. We denote the m+1...m+n −1 ideal generated by them as IM (q ). Let’s deﬁne:

D1...m = (−q)−l(σ)x ...x 1...m def σ∈Sm 1σ(1) mσ(m) P Dm+1...m+n = (−q)l(σ)x ...x . m+1...m+n def σ∈Sn m+1,m+σ(1) m+n,m+σ(n) P 1...m m+1...m+n D1...m and Dm+1...m+n represent respectively the quantum determinants of the quantum matrix bialgebras:

1...m Mq(1, m) = kq /IM (q), 1 ≤ i, j ≤ m,

m ...m n − +1 + −1 Mq 1 (m +1, m + n) = kq /IM (q ), m +1 ≤ k,l ≤ m + n.

m ...m n 1...m +1 + − D1...m and Dm+1...m+n are central elements in Mq(1, m) and Mq 1 (m +1, m + n) respectively ([PW] pg 50).

13 Further properties of these algebras and their quantum determinants are studied in [PW] ch. 5. We deﬁne:

1...m−1 1...m m...m−1 GLq(1, m) = Mq(1, m) < D1...m >/(D1...mD1...m − 1),

m ...m n−1 − +1 + Mq 1 (m +1, m + n) < Dm+1...m+n > GLq−1 (m +1, m + n) = m+1...m+n m+1...m+n−1 (Dm+1...m+nDm+1...m+n − 1). q q These are Hopf algebras, their antipodes S11, S22 are explicitly calculated in [PW] pg 57. Deﬁnition (4.6). Quantum general linear supergroup.

1...m−1 m+1...m+n−1 kq[GL(m|n)] =def kq[M(m|n)] < D1...m , Dm+1...m+n >

1...m−1 m+1...m+n−1 where D1...m , Dm+1...m+n are even indeterminates such that:

1...m 1...m−1 1...m−1 1...m D1...mD1...m = 1 = D1...m D1...m,

m+1...m+n m+1...m+n−1 m+1...m+n−1 m+1...m+n Dm+1...m+nDm+1...m+n = 1 = Dm+1...m+n Dm+1...m+n where X11 = (xij)1≤i,j≤m, X22 = (xij)m+1≤i,j≤m+n are matrices of even indeterminates and Ξ12 = (ξkl)1≤k≤m,m+1≤l≤m+n, Ξ12 = (ξkl)m+1≤k≤m+n,1≤l≤m are matrices of odd indeterminates. The xij’s, ξkl’s are the generators of kq[M(m|n)].

detq(M) denotes the determinant of the Manin matrix M (that is a matrix of indeterminates satisfying the Manin commutation relations). One can verify ([PW] ch. 5) that q m+1...m+n−1 detq(S22(X22)) = Dm+1...m+n . By an abuse of language we will denote with the same letters xij, ξkl and aij the indeterminates generators of the rings kq[M(m|n)] and kq[GL(m|n)] the context making clear where the elements sit.

We also deﬁne:

q q Berq =def detq(S22(X22))detq(X11 − Ξ12S22(X22)Ξ21). called the quantum berezinian. Berq is invertible in kq[GL(m|n)], the proof is a small variation of the one in (3.2).

We now want to explicitly give the coalgebra structure for the ring kq[GL(m|n)]. We need some lemmas on commutation relations.

14 Lemma (4.7).

1...m −1 1...m 1...m−1 1...m−1 1) D1...mξij = q ξijD1...m, D1...m ξij = qξijD1...m ,

m+1...m+n −1 m+1...m+n m+1...m+n−1 m+1...m+n−1 2) Dm+1...m+nξij = q ξijDm+1...m+n, Dm+1...m+n ξij = qξijDm+1...m+n , where 1 ≤ i ≤ m, m +1 ≤ j ≤ m + n or m +1 ≤ i ≤ m + n, 1 ≤ j ≤ m. Proof. Let’s prove (1). It is enough to prove the ﬁrst commutation relation. We will ﬁrst do for 1 ≤ i ≤ m, m +1 ≤ j ≤ m + n. By induction on m. For m = 1 it is a direct simple check. Assume for now that 1 ≤ i < m, m +1 ≤ j ≤ m + n (the case i = m is treated separately). Let D1...α...mˆ denote the quantum determinant of the quantum minor obtained from 1...β...mˆ the quantum matrix X11 by suppressing row α and column β. By [PW] formula at pg 47 on Laplace expansion of quantum determinants we have:

1...m m s−m 1...s...mˆ D1...mξij = s=1(−q) D1...m−1 xmsξij = P m s−m 1...s...mˆ = s=1(−q) D1...m−1 ξijxms. P By induction 1...s...mˆ −1 1...s...mˆ D1...m−1 ξij = q ξijD1...m−1 hence we have our result. Now the case i = m, m +1 ≤ j ≤ m + n. By [PW] formula at pg 47 we have:

1...m m s−1 1...s...mˆ D1...mξmj = s=1(−q) D2...m x1sξmj = P m s−1 1...s...mˆ −1 = s=1(−q) D2...m [ξmjx1s +(q − q)ξ1jxms]. P By induction: 1...s...mˆ −1 1...s...mˆ D2...m ξmj = q ξmjD2...m . So we have:

1...m −1 1...m −1 m s−1 1...s...mˆ D1...mξmj = q ξmjD1...m +(q − q) s=1(−q) D2...m xmsξ1j]. P But notice that ([PW] pg 47):

m s−1 1...s...mˆ 1...m (−q) D2...m xms = δ1mD1...m =0 Xs=1 hence we have our result. The case m +1 ≤ i ≤ m + n, 1 ≤ j ≤ m is done in a similar way.

15 The proof of (2) goes along the same lines. Lemma (4.8).

1...m −t 1...m 1.a) D1...ma1,k1 ...am,km = q a1,k1 ...am,km D1...m

1...m −t 1...m 1.b) D1...mak1,1 ...akm,m = q ak1,1 ...akm,mD1...m

m+1...m+n −s m+1...m+n 2.a) Dm+1...m+nam+1,l1 ...am+n,ln = q am+1,l1 ...am+n,ln Dm+1...m+n

m+1...m+n −s m+1...m+n 2.b) Dm+1...m+nal1,m+1 ...aln,m+n = q al1,m+1 ...aln,m+nDm+1...m+n

where t is such that m − t = |{k1 ...km}∩{1 ...m}|, 1 ≤ k1 <...

1...m ai,kj D1...m for 1 ≤ kj ≤ m 1...m D1...mai,kj =   −1 1...m q ai,kj D1...m otherwise  Hence we have the result. The proofs of (1.b) and (2.a), (2.b) are the same. As in the supercommutative case we can write:

m 1...m ∆(D1...m) = Ri Xi=1 with R = (−q)−l(σ)a ...a ⊗ a ...a i σ∈Sm,(k1,...,km)∈ρi 1,k1 m,km k1,σ(1) km,σ(m) P ρi = {(k1,...,km)|1 ≤ k1,...,km ≤ m + n, |{k1,...,km}∩{1 ...m}| = m − i} Similarly one can write: n m+1...m+n ∆(Dm+1...m+n) = Si Xi=1 with s = (−q)l(θ)a ...a ⊗ a ...a i θ∈Sn,(l1,...,ln)∈σi 1,l1 n,ln l1,θ(1) ln,θ(n) P σi = {(l1,...,ln)|1 ≤ l1,...,ln ≤ m + n, |{l1,...,ln}∩{m +1 ...m + n}| = n − i}.

16 Lemma (4.9). −2i −1 2i −1 1) R0Ri = q RiR0,R0 Ri = q RiR0 −2i −1 2i −1 2) S0Si = q SiS0, S0 Si = q SiS0 1...m 1...m Proof. Immediate from lemma (4.8) noting that R0 = D1...m ⊗ D1...m and S0 = m+1...m+n m+1...m+n Dm+1...m+n ⊗ Dm+1...m+n. Lemma (4.10). −2 −2m 1. R0(R1 + ... + Rm)=(q R1 + ... + q Rm)R0 −1 −1 −2 −2m 2. (R1 + ... + Rm)R0 = R0 (q R1 + ... + q Rm) −2 −2n 3. S0(S1 + ... + Sn)=(q S1 + ... + q Sn)S0 −1 −1 −2 −2n 4. (S1 + ... + Sn)S0 = S0 (q S1 + ... + q Sn) Proof. This is an immediate application of lemma (4.9).

Lemma (4.11). Let a s s ...a s s ∈ k [GL(m|n)], for all s, 1 ≤ s ≤ 2mn +1. i1,j1 im,jm q s s Assume that for all s at least one (ik,jk) is such that 1 ≤ ik ≤ m, m ≤ jk ≤ m + n or s m +1 ≤ ik ≤ m + n, 1 ≤ jk ≤ m, that is aik,jk is odd. Then

2mn+1 a s s ...a s s =0 i1,j1 im,jm sY=1

Proof. Let i be the map:

i 1...m−1 m+1...m+n−1 < kq[M(m|n)] ֒→ kq[M(m|n)] < D1...m , Dm+1...m+n

Let X ∈ kq[GL(m|n)] such that there exists a X0 ∈ kq[M(m|n)], i(X0) = X. If one wants to prove X = 0 it is enough to show X0 = 0. Now let 2mn+1 X = a s s ...a s s . i1,j1 im,jm sY=1 2mn+1 If X = a s s ...a s s ∈ k [M(m|n)] we have that i(X ) = X. 0 s=1 i1,j1 im,jm q 0 By theQ previous argument it is enough to show that X0 = 0.

By [Ma3] pg 172 we have that {1,ai1,j1 ...air ,jr }{(i1,j1)≤...≤(ir ,jr ),r≥1} form a basis for kq[M(m|n)], where ≤ is a suitable ordering on the indeces (ik,jk). Since the Manin relations are homogeneous we have:

X0 = c(r1,s1)...(rt,st)ar1,s1 ...art,st (⋆) r ,s ... r ,s ,c k ( 1 1)≤ ≤( t tX) (r1,s1)...(rt,st)∈ q where the sum is taken over a suitable set of ((r1,s1) ... (rt,st)) with t = m(2mn + 1).

17 By hypothesis each term in the sum (⋆) contains 2mn + 1 odd indeterminates and since the indeces are ordered, it is 0.

Proposition (4.12). Coalgebra structure for kq[GL(m|n)].

kq[GL(m|n)] is a coalgebra with comultiplication:

∆(aij) = aik ⊗ akl P

1...m−1 2mn+2 i−1 −1 −1 −2 −2m i−1 ∆(D1...m ) = i=1 (−1) R0 [R0 (q R1 + ... + q Rm)] P

m+1...m+n−1 2mn+2 i−1 −1 −1 −2 −2n i−1 ∆(Dm+1...m+n ) = i=1 (−1) S0 [S0 (q S1 + ... + q Sn)] P and counit:

1...m−1 m+1...m+n−1 ǫ(aij) = δij , ǫ(D1...m )=1, ǫ(Dm+1...m+n )=1

Proof. ∆ is well deﬁned on all the commutation relations among the generators. We only need to check that

1...m 1...m−1 1...m−1 1...m ∆(D1...m)∆(D1...m )=1=∆(D1...m )∆(D1...m),

m+1...m+n m+1...m+n−1 m+1...m+n−1 m+1...m+n ∆(Dm+1...m+n)∆(Dm+1...m+n )=1=∆(Dm+1...m+n )∆(Dm+1...m+n)

Let’s check the ﬁrst one.

1...m 1...m−1 ∆(D1...m)∆(D1...m ) =

−1 −2 −2 −2m [R0 +(R1 + ... + Rm)][R0 − R0 (q R1 + ... + q Rm) + ...] =

−1 −1 −2 −2m =1 ⊗ 1+(R1 + ...Rm)R0 − R0 (q R1 + ... + q Rm)+

−2 −2 −2m −(R1 + ... + Rm)R0 (q R1 + ... + q Rm) + ...

18 We obtain a telescopic sum. In fact, let’s see the generic terms:

i−1 −1 −1 −2 −2m i−1 G =[R0 +(R1 + ... + Rm)][... +(−1) R0 [R0 (q R1 + ... + q Rm)] +

i −1 −1 −2 −2m i +(−1) R0 [R0 (q R1 + ... + q Rm)] + ...] =

i−1 −1 −2 −2m i−1 = ... +(−1) [R0 (q R1 + ... + q Rm)] +

i−1 −1 −1 −2 −2m i−1 +(−1) (R1 + ... + Rm)R0 [R0 (q R1 + ... + q Rm)] +

i −1 −2 −2m i +(−1) [R0 (q R1 + ... + q Rm)] +

i −1 −1 −2 −2m i +(−1) (R1 + ... + Rm)R0 [R0 (q R1 + ... + q Rm)] .

−1 −1 −2 −2m By lemma (4.10)(2) we have that (R1 + ... + Rm)R0 = R0 (q R1 + ... + q Rm) so the second and third term in the sum G cancel each other.

The last term of the sum is 0 by lemma (4.11).

This completes the check for the ﬁrst part of the ﬁrst relation.

Let’s see the second part.

1...m−1 1...m ∆(D1...m )∆(D1...m) =

−1 −2 −2 −2m =[R0 − R0 (q R1 + ... + q Rm) + ...][R0 +(R1 + ... + Rm)] =

−1 −2 −2 −2m =1 ⊗ 1 + R0 (R1 + ...Rm) − R0 (q R1 + ... + q Rm)R0+

−2 −2 −2m −R0 (q R1 + ... + q Rm)(R1 + ... + Rm) + ...

By lemma (4.10)(1) the second and third term cancel out. Let’s see as before the generic

19 terms G′: ′ i−1 −1 −1 −2 −2m i−1 G =[... +(−1) R0 [R0 (q R1 + ... + q Rm)] +

i −1 −1 −2 −2m i +(−1) R0 [R0 (q R1 + ... + q Rm)] + ...][R0 +(R1 + ... + Rm)] =

i−1 −1 −1 −2 −2m i−1 = ... +(−1) R0 [R0 (q R1 + ... + q Rm)] R0+

i−1 −1 −1 −2 −2m i−1 +(−1) R0 [R0 (q R1 + ... + q Rm)] (R1 + ... + Rm)+

i −1 −1 −2 −2m i +(−1) R0 [R0 (q R1 + ... + q Rm)] R0+

i −1 −1 −2 −2m i +(−1) R0 [R0 (q R1 + ... + q Rm)] (R1 + ... + Rm) + ... We now look at the second and third term in G′. i−1 −1 −1 −2 −2m i−1 +(−1) R0 [R0 (q R1 + ... + q Rm)] (R1 + ... + Rm)+

i −1 −1 −2 −2m i−1 −1 −2 −2m +(−1) R0 [R0 (q R1 + ... + q Rm)] R0 (q R1 + ... + q Rm)R0. −2 −2m By lemma (4.10)(1) we have that (q R1 + ... + q Rm)R0 = R0(R1 + ... + Rm), so the second and third term cancel each other. The last term of the sum is 0 by lemma (4.11). The second relation can be checked in the same way. m|n m|n∗ Proposition (4.13). kq[GL(m|n)] admits a coaction on kq , kq . Proof. Immediate.

Now we need to give the antipode on kq[GL(m|n)]. q Proposition (4.14). kq[GL(m|n)] is an Hopf algebra with antipode S given by: q q q q q q S (X11) = S11(X11) + S11(X11)Ξ12S2(X22 − Ξ21S11(X11)Ξ12)Ξ21S11(X11)

q q q q S (Ξ12) = −S11(X11)Ξ12S2 (X22 − Ξ21S11(X11)Ξ12)

q q q q S (Ξ21) = −S2 (X22 − Ξ21S11(X11)Ξ12)Ξ21S11(X11)

q q q S (X22) = S2 (X22 − Ξ21S11(X11)Ξ12)

20 q 1...m−1 m+1...m+n S (D1...m ) = Dm+1...m+n

q m+1...m+n−1 1...m S (Dm+1...m+n ) = D1...m q q where X22 − Ξ21S11(X11)Ξ12 is a quantum matrix (see [P] §4) and S2 denotes its quantum antipode. Proof. See [P] §4.

Remark (4.15). In the ring kq[GL(m|n)] the parameter q can be specialized to any × value in k . Hence in the deﬁnition of kq[GL(m|n)] q can be also taken as any element in k×.

Example (4.16): kq[GL(1|1)].

−1 −1 kq[GL(1|1)] = k /I

where I is the ideal generated by the relations:

−1 −1 x11ξ12 = q ξ12x11, x11ξ21 = q ξ21x11,

ξ21x22 = qx22ξ21, ξ12x22 = qx22ξ12,

−1 x11x22 − x22x11 =(q − q )ξ12ξ21

2 2 ξ12ξ21 = −ξ21ξ12, ξ12 = ξ21 =0.

The coalgebra structure is the following.

∆(aij) = k aik ⊗ akj P −1 −1 −1 −2 −2 −2 ∆(x11 ) = x11 ⊗ x11 − q x11 ξ12 ⊗ x11 ξ21,

−1 −1 −1 2 −2 −2 ∆(x22 ) = x22 ⊗ x22 − q x22 ξ21 ⊗ x22 ξ12,

−1 −1 x11 ξ12 1 0 ǫ(x11 ) = ǫ(x22 )=1, ǫ = , ξ21 x22 0 1

−1 −1 −1 −1 −1 x11 ξ12 (x11 − ξ12x22 ξ21) −x11 ξ12(x22 − ξ21x11 ξ12) Sq   =   ξ x −x−1ξ (x − ξ x−1ξ )−1 (x − ξ x−1ξ )−1  21 22   22 21 11 12 22 21 22 21 11 12  −1 S(x11 ) = x22

−1 S(x22 ) = x11

21 kq[GL(1|1)] admits a coaction on the quantum spaces:

1|1 −1 kq = k/(xξ − q ξx)

1|1 ∗ −1 (kq ) = k/(yη − q ηy)

In this particular case is it immediate to construct also a deformation for the Hopf algebra k[SL(1|1)]. In fact since the quantum berezinian:

−1 −1 Berq = x22 (x11 − ξ12x22 ξ21)

is a central element in kq[M(m|n)] and in kq[GL(m|n)] we can deﬁne:

−1 −1 ′ kq[SL(1|1)] = k /I

where I′ is the two-sided ideal generated by the same relations as I together with the extra

relation Berq = 1.

The comultiplication, counit, antipode are the same as kq[GL(1|1)], in fact one can check directly that they are still well deﬁned.

We plan to construct in a forthcoming paper the deformation kq[SL(m|n)] and its

relation with kq[GL(m|n)].

REFERENCES

[Be] Berezin, F. A., Introduction to superanalysis, Kluwer Academic Publisher (1987). [Bo] Borel, A., Linear algebraic groups, GTM Springer-Verlag, (1991). [CP] Chari, V., Pressley, A., A guide to quantum groups, Cambridge University Press, (1994). [DM] Deligne, P., Morgan, J. Supersymmetry, Quantum Fields and strings. A course for mathematicians, Vol 1, AMS, (1999). [Fi] Fioresi, R., Supergroups, quantum supergroups and their homogeneous spaces, Modern Physics Letters A, 16, no. 4-6, 269-274, (2001). [Ha] Hartshorne, R., Algebraic geometry, GTM Springer-Verlag, (1977). [Ko] Kostant, B., Graded manifolds, graded Lie theory and prequantization, Springer LNM 570, (1977). [Le] Leites, D. A., Introduction to the theory of supermanifolds, Russian Math. Surveys, 35, no. 1, 1-64, (1980). [LS] Lyubashenko V., Sudbery A., Quantum supergroups of GL(m|n) type, Duke Math. J., 90, no. 1, (1997).

22 [Ma1] Manin, Yu., Gauge ﬁeld theory and complex geometry, Springer Verlag, (1988). [Ma2] Manin, Yu., Topics in non commutative geometry, Princeton University Press, (1991). [Ma3] Manin, Yu., Multiparametric quantum deformation of the general linear supergroup, Comm. Math. Phy., 123, 163-175, (1989). [Ma4] Manin, Yu., Quantum groups and non commutative geometry, Centre de Reserches Mathematiques Montreal, 49, (1988). [Mo] Montgomery, S. Hopf algebras and their actions on rings, AMS, Providence, (1993).

[P] Phung, Ho Hai, On the structure of the quantum supergroups GLq(m|n), q-alg/9511- 23, (1999). [PW] Parshall, B.; Wang, J. P., Quantum linear groups, Mem. Amer. Math. Soc. 89, no. 439, (1991).